Mathematics

We settle the fusion product decomposition theorem for higher-level affine Demazure modules for the cases E (1) 6,7,8 , F (1) 4 and E (2) 6 , thus completing the main theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory, 2016). We obtain a new combinatorial proof for the key fact, that was used in Chari et al. (op cit.), to prove this decomposition theorem. We give a case free uniform proof for this key fact.

A linear étale representation of a complex algebraic group G is given by a complex algebraic G -module V such that G has a Zariski-open orbit on V and dimG=dimV . A current line of research investigates which étale representations can occur for reductive algebraic groups. Since a complete classification seems out of reach, it is of interest to find new examples of étale representations for such groups. The aim of this note is to describe two classical constructions of Vinberg and of Bala & Carter for nilpotent orbit classifications in semisimple Lie algebras, and to determine which reductive groups and étale representations arise in these constructions. We also explain in detail the relation between these two~constructions.

We study a problem concerning parabolic induction in certain p -adic unitary groups. More precisely, for E/F a quadratic extension of p -adic fields the associated unitary group G=U(n,n+1) contains a parabolic subgroup P with Levi component L isomorphic to GL n (E)? U 1 (E) . Let ? be an irreducible supercuspidal representation of L of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation ι G P ? is reducible.

Let G be a real linear algebraic group and K be a maximal compact subgroup. Let p be a standard minimal parabolic subalgebra of the complexified Lie algebra g of G . In this paper we show that: for any moderate growth smooth Fréchet representation V of G , the map V K ?�V induces isomorphisms H i (n, V K )??H i (n,V) ( ?�i?? ). This is called Casselman's comparison theorem in the literature. As a consequence, we show that: for any k?? , n k V is a closed subspace of V and the inclusion V K ?�V induces an isomorphism V K / n k V K =V/ n k V . This strengthens Casselman's automatic continuity theorem.

We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for E/F a quadratic extension of p-adic fields the associated unitary group G=U(n,n) contains a parabolic subgroup P with Levi component L isomorphic to GL n (E) . Let π be an irreducible supercuspidal representation of L of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation ι G P π is reducible.

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal meets a unique largest nilpotent orbit in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals with the same largest nilpotent orbit. We include two applications of the result, one to the higher vanishing of cohomology groups of vector bundles on the flag variety and another to the Kazhdan-Lusztig cells in the affine Weyl group of the symmetric group. Finally, some combinatorial results are discussed.

We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let (C,E,s) be an extriangulated category with enough projectives P and M be a full subcategory of C containing P . We show that certain quotient category of $\mathfrak{s}\textup{-def}(\mathcal{M})$, the category of s -deflations f: M 1 → M 2 with M 1 , M 2 ∈M , is abelian. Our main theorem has two applications. If M=C , we obtain that certain ideal quotient category $\mathfrak{s}\textup{-tri}(\mathcal{C})/\mathcal{R}_2$ is equivalent to the category of finitely presented modules $\textup{mod-}\mathcal{C}/[\mathcal{P}]$, where s -tri (C) is the category of all s -triangles. If M is a rigid subcategory, we show that $\mathcal{M}_{L}/[\mathcal{M}]\cong\textup{mod-}(\mathcal{M}/[\mathcal{P}])$ and $\mathcal{M}_{L}/[\Omega\mathcal{M}]\cong(\textup{mod-}(\mathcal{M}/[\mathcal{P}])^{\textup{op}})^{\textup{op}}$, where M L (resp. ΩM ) is the full subcategory of C of objects X admitting an s -triangle $\xymatrixrowsep{0.1pc}\xymatrix{X\ar[r]&M_1\ar[r] & M_2\ar@{-->}[r]&} (\textup{resp.} \xymatrixrowsep{0.1pc}\xymatrix{X\ar[r]&P\ar[r] & M\ar@{-->}[r]&})$ with M 1 , M 2 ∈M (resp. M∈M and P∈P ). In particular, we have $\mathcal{C}/[\mathcal{M}]\cong\textup{mod-}(\mathcal{M}/[\mathcal{P}])$ and $\mathcal{C}/[\Omega\mathcal{M}]\cong(\textup{mod-}(\mathcal{M}/[\mathcal{P}])^{\textup{op}})^{\textup{op}}$ provided that M is a cluster-tilting subcategory.

This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible representations. Only at the end we establish a new result: a natural bijection between the set of irreducible representations of an affine Hecke algebra with real parameters ≥1 , and the set of irreducible representations of the affine Weyl group underlying the algebra. This can be regarded as a generalized Springer correspondence with affine Hecke algebras.

In this paper, we study the affine Springer fiber F l N in type A for rectangular type semisimple nil-element N and calculate the relative position between irreducible components. In particular, we use the affine matrix ball construction to show the relative position map is compatible with the Kazhdan-Lusztig cell structure, generalizing the work of Steinberg and van Leeuwen.

The parabolic category O for affine gl N at level −N−e admits a structure of a categorical representation of sl ˜ e with respect to some endofunctors E and F . This category contains a smaller category A that categorifies the higher level Fock space. We prove that the functors E and F in the category A are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor F for the category A at level −N−e can be decomposed in terms of the components of the functor F for the category A at level −N−e−1 . To prove this, we use the following fact: a category with an action of sl ˜ e+1 contains a (canonically defined) subcategory with an action of sl ˜ e . We also prove a general statement that says that in some general situation a functor that satisfies a list of axioms is automatically Koszul dual to some sort of Zuckerman functor.